Given that 3^9 + 3^12 + 3^15 + 3^n is a perfect cube number. Calculate the
value of n.
3^9 + 3^12 + 3^15 + 3^n
= 3^9(1 + 3^3 + 3^6 + 3^(n-9) )
= 3^9(757 + 3^(n-9))
3^9 is a perfect cube, so all we need is
(757 + 3^(n-9)) to be a perfect cube, clearly n >9
perfect cubes past 757 with 3^(n-9) as the 2nd number:
1000 243 <---- well, that was lucky
1331 574
...
we know that 243 = 3^5
so n-9 = 5
n = 14
check:
if n=14
3^9 + 3^12 + 3^15 + 3^n
= 3^9 + 3^12 + 3^15 + 3^14
= 3^9 (1 + 3^3 + 3^6 + 3^5)
= 3^9 ( (1 + 27 + 729 + 243)
= 3^9 * 1000
= 3^9 * 10^3
= (3^3)^3 * 10^3
= 270^3
which is a perfect cube
To find the value of n in the equation 3^9 + 3^12 + 3^15 + 3^n is a perfect cube number, we need to use the properties of perfect cubes.
First, recall that a perfect cube is a number that can be expressed as the cube of an integer. For example, 8 is a perfect cube because it can be written as 2^3 (2 multiplied by itself three times).
Now, let's simplify the given expression. We have:
3^9 + 3^12 + 3^15 + 3^n
Notice that all the terms have a common base of 3. We can use this to our advantage to factor out 3^9:
3^9 (1 + 3^3 + 3^6 + 3^(n-9))
Now, we want this expression to be a perfect cube. Let's focus on the remaining expression inside the parentheses:
1 + 3^3 + 3^6 + 3^(n-9)
Here, we don't know the exact value of n yet, but we know that it must be an integer to maintain the pattern of perfect cubes.
To make this expression a perfect cube, we need to find a value of n where the sum of the terms is a perfect cube. We can start by testing some values of n to see if they satisfy this condition.
Let's try n = 12:
1 + 3^3 + 3^6 + 3^(12-9) = 1 + 27 + 729 + 3^3 = 1 + 27 + 729 + 27 = 784
We can see that 784 is not a perfect cube.
Now, let's try n = 15:
1 + 3^3 + 3^6 + 3^(15-9) = 1 + 27 + 729 + 3^6 = 1 + 27 + 729 + 729 = 1486
Again, 1486 is not a perfect cube.
Let's continue testing values of n until we find a perfect cube.
For n = 18:
1 + 3^3 + 3^6 + 3^(18-9) = 1 + 27 + 729 + 3^9 = 1 + 27 + 729 + 19683 = 20440
As you can see, 20440 is not a perfect cube either.
We can conclude that there is no integer value of n that makes the expression 3^9 + 3^12 + 3^15 + 3^n a perfect cube number.
Therefore, the value of n that satisfies this condition is undefined or nonexistent.
or be chall haat
Fhuthhhj
3^36 + ? = perfect cube
? Has to equal 3^?
Am I making sense?
See if you can figure it out now
(☆^ー^☆)